Let us consider the vertex functions, with \(Q_{1(2)}\)  the electric quadrupole moments with
indices suppressed, which contribute to the tail terms shown in Figure REF  and the graviton
self–interaction vertex. The symbols \(h_{1(2)}\) 
denote the respective field couplings given by \(O_k(\Phi _{1(2)})\)  as different linear differential
operators
acting on the decomposition of the gravitational field into 10 fields \(\Phi \) , as described in [1]}.
One obtains
\(V_{Qh}~&=&~ Q_1 h_1 - Q_2 h_2 ~=~ Q_- h_+ + Q_+ h_-,\\V_{Qh^2}~&=&~Q_1 h_1^2 - Q_2 h_2^2 ~=~ \frac{1}{\sqrt{2}} \left[2 h_+ h_- Q_+ + (h_+^2 + h_-^2) Q_-\right],\\V_{h^3}~&=&~\left[h_1^3 - h_2^3 \right] ~=~ \frac{1}{\sqrt{2}}\left[ 3 h_+^2 h_- + h_-^3\right].\) 
